Spoiler:
Assume L is infinite (the finite case is easier). Thus, L has an enumerator f.
Suppose we have a TM that on input x gets some element of Im(f) and, does something with x and this element, and accepts/rejects based on the outcome.
Now, back to what we wanted to prove, assume towards contradiction that for every i, L and Ai intersects.
As they intersect, L must have an element of Ai, so it is a possible outcome of the enumerator…
Think of how to use all this to obtain a contradiction.

If we consider the case that L is all TMs that always halts, how is it possible that an intersection between any Ai and L will be empty? Ai contains all TMs that their language is Li, including deciders, which are TMs that always halts.